Skip to main content
Springer Nature Link
Log in

Feedback control of nonlinear differential algebraic systems using Hamiltonian function method

  • Published:
Science in China Series F: Information Sciences Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

The stabilization and H ∞ control of nonlinear differential algebraic systems (NDAS) are investigated using the Hamiltonian function method. Firstly, we put forward a novel dissipative Hamiltonian realization (DHR) structure and give the condition to complete the Hamiltonian realization. Then, based on the DHR, we present a criterion for the stability analysis of NDAS and construct a stabilization controller for NDAS in absence of disturbances. Finally, for NDAS in presence of disturbances, the L 2 gain is analyzed via generalized Hamilton-Jacobi inequality and an H ∞ control strategy is constructed. The proposed stabilization and robust controller can effectively take advantage of the structural characteristics of NDAS and is simple in form.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Venkatasubramanian V, Schättler H, Zaborszky J. Dynamics of large constrained nonlinear systems—a taxonomy theory. Proc IEEE, 1995, 83(11): 1530–1561

    Article  Google Scholar 

  2. Hill D J, Mareels I M. Y. Stability theory for differential algebraic systems with application to power systems. IEEE Trans Circuits Syst, 1990, 37(11): 1416–1423

    Article  MATH  MathSciNet  Google Scholar 

  3. McClamboch N H, Wang D W. Feedback stabilization and tracking of constrained robots. IEEE Trans Autom Control, 1988, 33(5): 419–426

    Article  Google Scholar 

  4. Campbell S L. A general method for nonlinear descriptor systems: A example from robotic path control. Proceedings of the 27th IEEE Conference on Decision and Control, Austin, Texas, Dec 1988, 630–631

  5. Zimmer G, Meier J. On observing nonlinear descriptor system. Syst Control Lett, 1997, 32: 43–48

    Article  MATH  MathSciNet  Google Scholar 

  6. Liu Y Q, Li Y Q. Stabilization of nonlinear singular systems. Proceedings of the American Control Conference, Philadelphia, Pennsylvania, Jun 1998, 2532–2533

  7. Wang J, Chen C, La Scala M. Parametric adaptive control of multimachine power systems with nonlinear loads. IEEE Trans Circuits Syst I-Regul Pap, 2004, 51(2): 91–100

    Article  Google Scholar 

  8. Wang H S, Yung C F, Chang F R. H∞ control for nonlinear descriptor systems. IEEE Trans Autom Control, 2002, 47(11): 1919–1925

    Article  MathSciNet  Google Scholar 

  9. Chiang H D, Fekih-Ahmed L. On the direct method for transient stability analysis of power system structure preserving models. Proceedings of 1992 IEEE International Symposium on Circuits and Systems, 1992, 5: 2545–2548

    Google Scholar 

  10. Praprost K L, Loparo K A. An energy function method for determining voltage collapse during a power system transient. IEEE Trans Circuits Syst I-Regul Pap, 1994, 41(10): 635–651

    Article  Google Scholar 

  11. Ortega R, Van der Schaft A J, Maschke B, et al. Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems, Automatica, 2002, 38(4): 585–596

    Article  MATH  MathSciNet  Google Scholar 

  12. Ortega R, Van der Schaft A J, Mareels I, et al. Putting energy back in control. IEEE Control Syst Mag, 2001, 21(2): 18–33

    Article  Google Scholar 

  13. Wang Y, Cheng D, Hong Y. Stabilization of synchronous generators with Hamiltonian function approach. Int J Syst Sci, 2001, 32(8): 971–978

    Article  MATH  MathSciNet  Google Scholar 

  14. Cheng D, Xi Z, Hong Y, et al. Energy-based stabilization in power systems. Proceedings of the 14th IFAC World Congress, Beijing, China, 1999, 297–303

  15. Sun Y, Shen T, Ortega R, et al. Decentralized controller design for multimachine power systems based on Hamiltonian structure. Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, 2001, 4: 3045–3050

    Google Scholar 

  16. Wang Y, Cheng D, Li C, et al. Dissipative Hamiltonian realization and energy-based L2 disturbance attenuation control of multi-machine power systems. IEEE Trans Autom Control, 2003, 48(8): 1428–1433

    Article  MathSciNet  Google Scholar 

  17. Bachamann R, Brüll L, Mrziglod T, et al. On methods for reducing the index of differential-algebraic equations. Comput Chem Eng, 1990, 14: 1271–1273

    Article  Google Scholar 

  18. Gear C W. Differential-algebraic equation index transformation. SIAM J Sci Stat Comp, 1988, 9: 39–47

    Article  MATH  MathSciNet  Google Scholar 

  19. Liu X P, Čelikovský S. Feedback control of affine nonlinear singular control systems. Int J Control, 1997, 68(4): 753–774

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Automation, Tsinghua University, Beijing, 100084, China

    Liu Yanhong & Li Chunwen

  2. Frick Laboratory, Princeton University, Princeton, NJ, 08544, USA

    Wu Rebing

Authors
  1. Liu Yanhong
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Li Chunwen
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Wu Rebing
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Liu Yanhong.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, Y., Li, C. & Wu, R. Feedback control of nonlinear differential algebraic systems using Hamiltonian function method. SCI CHINA SER F 49, 436–445 (2006). https://doi.org/10.1007/s11432-006-2004-8

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11432-006-2004-8

Keywords